Lesson 19: The Substitution Technique; Applications of Integration

Here's the on-ramp for this lesson.

What Will We Learn?

• This lesson is divided into two main chunks. The first introduces one of the most widely used integration techniques. Formally, it's called The Substitution Technique. Informally, it's known as "u-substitution" because of the way in which it's used in practice. The Substitution Technique is like the integral version of the Chain Rule -- whereas the latter teaches us how to differentiate a composite function, the former teaches us how to integrate an integrand that came from a Chain Rule application. (Recall that derivatives and integrals undo each other.) The second chunk of the lesson focuses on the applications of integration. Unlike differentiation, where we could (and did) talk about applications early on in the development of the topic, most of the applications of integration involve actually calculating integrals, which meant that we needed to wait to discuss those applications until we learned how to calculate enough integrals. That is why this topic is at the end of the chapter, and indeed ends the chapter. After this lesson we will therefore have completed our study of calculus (woohoo!) using only algebraic functions. Starting in the next lesson we will go back to the beginning and revisit all the greatest hits of calculus, this time focusing on transcendental functions.

Why Do We Need to Learn This?

• The Substitution Technique is the go-to integration technique. It is the first thing one tries -- after, of course, trying to use an established rule/result like the integral version of the Power Rule. And in many cases, it works well. In a Calculus 2 course additional integration techniques are discussed that come in handy when $u$-substitution cannot be applied.